Calculus Examples

$(x^{2} - 11 x + 32) e^{x}$

Step 1

Write $(x^{2} - 11 x + 32) e^{x}$ as a function.

$f (x) = (x^{2} - 11 x + 32) e^{x}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} - 11 x + 32$ and $g (x) = e^{x}$ .

$(x^{2} - 11 x + 32) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2} - 11 x + 32]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} \frac{d}{d x} [x^{2} - 11 x + 32]$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

By the Sum Rule, the derivative of $x^{2} - 11 x + 32$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [32]$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [32])$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (2 x + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [32])$

Step 2.1.3.3

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (2 x - 11 \frac{d}{d x} [x] + \frac{d}{d x} [32])$

Step 2.1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (2 x - 11 \cdot 1 + \frac{d}{d x} [32])$

Step 2.1.3.5

Multiply $- 11$ by $1$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (2 x - 11 + \frac{d}{d x} [32])$

Step 2.1.3.6

Since $32$ is constant with respect to $x$ , the derivative of $32$ with respect to $x$ is $0$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (2 x - 11 + 0)$

Step 2.1.3.7

Add $2 x - 11$ and $0$ .

$(x^{2} - 11 x + 32) e^{x} + e^{x} (2 x - 11)$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Apply the distributive property.

$x^{2} e^{x} - 11 x e^{x} + 32 e^{x} + e^{x} (2 x - 11)$

Step 2.1.4.2

Apply the distributive property.

$x^{2} e^{x} - 11 x e^{x} + 32 e^{x} + e^{x} (2 x) + e^{x} \cdot - 11$

Step 2.1.4.3

Combine terms.

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Step 2.1.4.3.1

Move $- 11$ to the left of $e^{x}$ .

$x^{2} e^{x} - 11 x e^{x} + 32 e^{x} + e^{x} (2 x) - 11 \cdot e^{x}$

Step 2.1.4.3.2

Add $- 11 x e^{x}$ and $e^{x} (2 x)$ .

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Step 2.1.4.3.2.1

Move $e^{x}$ .

$x^{2} e^{x} - 11 x e^{x} + 2 x e^{x} + 32 e^{x} - 11 e^{x}$

Step 2.1.4.3.2.2

Add $- 11 x e^{x}$ and $2 x e^{x}$ .

$x^{2} e^{x} - 9 x e^{x} + 32 e^{x} - 11 e^{x}$

Step 2.1.4.3.3

Subtract $11 e^{x}$ from $32 e^{x}$ .

$x^{2} e^{x} - 9 x e^{x} + 21 e^{x}$

Step 2.1.4.4

Reorder terms.

$e^{x} x^{2} - 9 e^{x} x + 21 e^{x}$

Step 2.1.4.5

Reorder factors in $e^{x} x^{2} - 9 e^{x} x + 21 e^{x}$ .

$f' (x) = x^{2} e^{x} - 9 x e^{x} + 21 e^{x}$

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} e^{x} - 9 x e^{x} + 21 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- 9 x e^{x}] + \frac{d}{d x} [21 e^{x}]$ .

$\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [- 9 x e^{x}] + \frac{d}{d x} [21 e^{x}]$

Step 2.2.2

Evaluate $\frac{d}{d x} [x^{2} e^{x}]$ .

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Step 2.2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{x}$ .

$x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x e^{x}] + \frac{d}{d x} [21 e^{x}]$

Step 2.2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x e^{x}] + \frac{d}{d x} [21 e^{x}]$

Step 2.2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} e^{x} + e^{x} (2 x) + \frac{d}{d x} [- 9 x e^{x}] + \frac{d}{d x} [21 e^{x}]$

Step 2.2.3

Evaluate $\frac{d}{d x} [- 9 x e^{x}]$ .

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Step 2.2.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x e^{x}$ with respect to $x$ is $- 9 \frac{d}{d x} [x e^{x}]$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 \frac{d}{d x} [x e^{x}] + \frac{d}{d x} [21 e^{x}]$

Step 2.2.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x}$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [21 e^{x}]$

Step 2.2.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 (x e^{x} + e^{x} \frac{d}{d x} [x]) + \frac{d}{d x} [21 e^{x}]$

Step 2.2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} [21 e^{x}]$

Step 2.2.3.5

Multiply $e^{x}$ by $1$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 (x e^{x} + e^{x}) + \frac{d}{d x} [21 e^{x}]$

Step 2.2.4

Evaluate $\frac{d}{d x} [21 e^{x}]$ .

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Step 2.2.4.1

Since $21$ is constant with respect to $x$ , the derivative of $21 e^{x}$ with respect to $x$ is $21 \frac{d}{d x} [e^{x}]$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 (x e^{x} + e^{x}) + 21 \frac{d}{d x} [e^{x}]$

Step 2.2.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$x^{2} e^{x} + e^{x} (2 x) - 9 (x e^{x} + e^{x}) + 21 e^{x}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Apply the distributive property.

$x^{2} e^{x} + e^{x} (2 x) - 9 (x e^{x}) - 9 e^{x} + 21 e^{x}$

Step 2.2.5.2

Combine terms.

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Step 2.2.5.2.1

Subtract $9 x e^{x}$ from $e^{x} (2 x)$ .

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Step 2.2.5.2.1.1

Move $e^{x}$ .

$x^{2} e^{x} + 2 x e^{x} - 9 x e^{x} - 9 e^{x} + 21 e^{x}$

Step 2.2.5.2.1.2

Subtract $9 x e^{x}$ from $2 x e^{x}$ .

$x^{2} e^{x} - 7 x e^{x} - 9 e^{x} + 21 e^{x}$

Step 2.2.5.2.2

Add $- 9 e^{x}$ and $21 e^{x}$ .

$x^{2} e^{x} - 7 x e^{x} + 12 e^{x}$

Step 2.2.5.3

Reorder terms.

$e^{x} x^{2} - 7 e^{x} x + 12 e^{x}$

Step 2.2.5.4

Reorder factors in $e^{x} x^{2} - 7 e^{x} x + 12 e^{x}$ .

$f'' (x) = x^{2} e^{x} - 7 x e^{x} + 12 e^{x}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $x^{2} e^{x} - 7 x e^{x} + 12 e^{x}$ .

$x^{2} e^{x} - 7 x e^{x} + 12 e^{x}$

Step 3

Set the second derivative equal to $0$ then solve the equation $x^{2} e^{x} - 7 x e^{x} + 12 e^{x} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$x^{2} e^{x} - 7 x e^{x} + 12 e^{x} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $e^{x}$ out of $x^{2} e^{x} - 7 x e^{x} + 12 e^{x}$ .

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Step 3.2.1.1

Factor $e^{x}$ out of $x^{2} e^{x}$ .

$e^{x} x^{2} - 7 x e^{x} + 12 e^{x} = 0$

Step 3.2.1.2

Factor $e^{x}$ out of $- 7 x e^{x}$ .

$e^{x} x^{2} + e^{x} (- 7 x) + 12 e^{x} = 0$

Step 3.2.1.3

Factor $e^{x}$ out of $12 e^{x}$ .

$e^{x} x^{2} + e^{x} (- 7 x) + e^{x} \cdot 12 = 0$

Step 3.2.1.4

Factor $e^{x}$ out of $e^{x} x^{2} + e^{x} (- 7 x)$ .

$e^{x} (x^{2} - 7 x) + e^{x} \cdot 12 = 0$

Step 3.2.1.5

Factor $e^{x}$ out of $e^{x} (x^{2} - 7 x) + e^{x} \cdot 12$ .

$e^{x} (x^{2} - 7 x + 12) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $x^{2} - 7 x + 12$ using the AC method.

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Step 3.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 3.2.2.1.2

Write the factored form using these integers.

$e^{x} ((x - 4) (x - 3)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$e^{x} (x - 4) (x - 3) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{x} = 0$

$x - 4 = 0$

$x - 3 = 0$

Step 3.4

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 3.4.2

Solve $e^{x} = 0$ for $x$ .

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Step 3.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 3.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 3.4.2.3

There is no solution for $e^{x} = 0$

No solution

Step 3.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.7

The final solution is all the values that make $e^{x} (x - 4) (x - 3) = 0$ true.

$x = 4, 3$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $4$ in $f (x) = (x^{2} - 11 x + 32) e^{x}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = ({(4)}^{2} - 11 \cdot 4 + 32) \cdot e^{4}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $4$ to the power of $2$ .

$f (4) = (16 - 11 \cdot 4 + 32) \cdot e^{4}$

Step 4.1.2.1.2

Multiply $- 11$ by $4$ .

$f (4) = (16 - 44 + 32) \cdot e^{4}$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Subtract $44$ from $16$ .

$f (4) = (- 28 + 32) \cdot e^{4}$

Step 4.1.2.2.2

Add $- 28$ and $32$ .

$f (4) = 4 e^{4}$

Step 4.1.2.3

The final answer is $4 e^{4}$ .

$4 e^{4}$

Step 4.2

The point found by substituting $4$ in $f (x) = (x^{2} - 11 x + 32) e^{x}$ is $(4, 4 e^{4})$ . This point can be an inflection point.

$(4, 4 e^{4})$

Step 4.3

Substitute $3$ in $f (x) = (x^{2} - 11 x + 32) e^{x}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = ({(3)}^{2} - 11 \cdot 3 + 32) \cdot e^{3}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $3$ to the power of $2$ .

$f (3) = (9 - 11 \cdot 3 + 32) \cdot e^{3}$

Step 4.3.2.1.2

Multiply $- 11$ by $3$ .

$f (3) = (9 - 33 + 32) \cdot e^{3}$

Step 4.3.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.1

Subtract $33$ from $9$ .

$f (3) = (- 24 + 32) \cdot e^{3}$

Step 4.3.2.2.2

Add $- 24$ and $32$ .

$f (3) = 8 e^{3}$

Step 4.3.2.3

The final answer is $8 e^{3}$ .

$8 e^{3}$

Step 4.4

The point found by substituting $3$ in $f (x) = (x^{2} - 11 x + 32) e^{x}$ is $(3, 8 e^{3})$ . This point can be an inflection point.

$(3, 8 e^{3})$

Step 4.5

Determine the points that could be inflection points.

$(4, 4 e^{4}), (3, 8 e^{3})$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 3) \cup (3, 4) \cup (4, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 3)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2.9$ in the expression.

$f'' (2.9) = {(2.9)}^{2} e^{2.9} - 7 \cdot 2.9 \cdot e^{2.9} + 12 e^{2.9}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $2.9$ to the power of $2$ .

$f'' (2.9) = 8.41 e^{2.9} - 7 \cdot 2.9 \cdot e^{2.9} + 12 e^{2.9}$

Step 6.2.1.2

Multiply $8.41$ by $e^{2.9}$ .

$f'' (2.9) = 152.84456255 - 7 \cdot 2.9 \cdot e^{2.9} + 12 e^{2.9}$

Step 6.2.1.3

Multiply $- 7$ by $2.9$ .

$f'' (2.9) = 152.84456255 - 20.3 \cdot e^{2.9} + 12 e^{2.9}$

Step 6.2.1.4

Multiply $- 20.3$ by $e^{2.9}$ .

$f'' (2.9) = 152.84456255 - 368.93515099 + 12 e^{2.9}$

Step 6.2.2

Simplify by adding terms.

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Step 6.2.2.1

Subtract $368.93515099$ from $152.84456255$ .

$f'' (2.9) = - 216.09058844 + 12 e^{2.9}$

Step 6.2.2.2

Add $- 216.09058844$ and $12 e^{2.9}$ .

$f'' (2.9) = 1.99915599$

Step 6.2.3

The final answer is $1.99915599$ .

$1.99915599$

Step 6.3

At $2.9$ , the second derivative is $1.99915599$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 3)$ .

Increasing on $(- \infty, 3)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(3, 4)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $\frac{7}{2}$ in the expression.

$f'' (\frac{7}{2}) = {(\frac{7}{2})}^{2} e^{\frac{7}{2}} - 7 (\frac{7}{2}) \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Apply the product rule to $\frac{7}{2}$ .

$f'' (\frac{7}{2}) = \frac{7^{2}}{2^{2}} \cdot e^{\frac{7}{2}} - 7 (\frac{7}{2}) \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.2

Raise $7$ to the power of $2$ .

$f'' (\frac{7}{2}) = \frac{49}{2^{2}} \cdot e^{\frac{7}{2}} - 7 (\frac{7}{2}) \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.3

Raise $2$ to the power of $2$ .

$f'' (\frac{7}{2}) = \frac{49}{4} \cdot e^{\frac{7}{2}} - 7 (\frac{7}{2}) \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.4

Combine $\frac{49}{4}$ and $e^{\frac{7}{2}}$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - 7 (\frac{7}{2}) \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.5

Multiply $- 7 (\frac{7}{2})$ .

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Step 7.2.1.5.1

Combine $- 7$ and $\frac{7}{2}$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} + \frac{- 7 \cdot 7}{2} \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.5.2

Multiply $- 7$ by $7$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} + \frac{- 49}{2} \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.6

Move the negative in front of the fraction.

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - \frac{49}{2} \cdot e^{\frac{7}{2}} + 12 e^{\frac{7}{2}}$

Step 7.2.1.7

Combine $e^{\frac{7}{2}}$ and $\frac{49}{2}$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - \frac{e^{\frac{7}{2}} \cdot 49}{2} + 12 e^{\frac{7}{2}}$

Step 7.2.1.8

Move $49$ to the left of $e^{\frac{7}{2}}$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - \frac{49 e^{\frac{7}{2}}}{2} + 12 e^{\frac{7}{2}}$

Step 7.2.2

To write $- \frac{49 e^{\frac{7}{2}}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - \frac{49 e^{\frac{7}{2}}}{2} \cdot \frac{2}{2} + 12 e^{\frac{7}{2}}$

Step 7.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.1

Multiply $\frac{49 e^{\frac{7}{2}}}{2}$ by $\frac{2}{2}$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - \frac{49 e^{\frac{7}{2}} \cdot 2}{2 \cdot 2} + 12 e^{\frac{7}{2}}$

Step 7.2.3.2

Multiply $2$ by $2$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}}}{4} - \frac{49 e^{\frac{7}{2}} \cdot 2}{4} + 12 e^{\frac{7}{2}}$

Step 7.2.4

Combine the numerators over the common denominator.

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}} - 49 e^{\frac{7}{2}} \cdot 2}{4} + 12 e^{\frac{7}{2}}$

Step 7.2.5

Simplify each term.

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Step 7.2.5.1

Simplify the numerator.

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Step 7.2.5.1.1

Multiply $2$ by $- 49$ .

$f'' (\frac{7}{2}) = \frac{49 e^{\frac{7}{2}} - 98 e^{\frac{7}{2}}}{4} + 12 e^{\frac{7}{2}}$

Step 7.2.5.1.2

Subtract $98 e^{\frac{7}{2}}$ from $49 e^{\frac{7}{2}}$ .

$f'' (\frac{7}{2}) = \frac{- 49 e^{\frac{7}{2}}}{4} + 12 e^{\frac{7}{2}}$

Step 7.2.5.2

Move the negative in front of the fraction.

$f'' (\frac{7}{2}) = - \frac{49 e^{\frac{7}{2}}}{4} + 12 e^{\frac{7}{2}}$

Step 7.2.6

To write $12 e^{\frac{7}{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f'' (\frac{7}{2}) = - \frac{49 e^{\frac{7}{2}}}{4} + 12 e^{\frac{7}{2}} \cdot \frac{4}{4}$

Step 7.2.7

Combine fractions.

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Step 7.2.7.1

Combine $12 e^{\frac{7}{2}}$ and $\frac{4}{4}$ .

$f'' (\frac{7}{2}) = - \frac{49 e^{\frac{7}{2}}}{4} + \frac{12 e^{\frac{7}{2}} \cdot 4}{4}$

Step 7.2.7.2

Combine the numerators over the common denominator.

$f'' (\frac{7}{2}) = \frac{- 49 e^{\frac{7}{2}} + 12 e^{\frac{7}{2}} \cdot 4}{4}$

Step 7.2.8

Simplify the numerator.

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Step 7.2.8.1

Multiply $4$ by $12$ .

$f'' (\frac{7}{2}) = \frac{- 49 e^{\frac{7}{2}} + 48 e^{\frac{7}{2}}}{4}$

Step 7.2.8.2

Add $- 49 e^{\frac{7}{2}}$ and $48 e^{\frac{7}{2}}$ .

$f'' (\frac{7}{2}) = \frac{- e^{\frac{7}{2}}}{4}$

Step 7.2.9

Move the negative in front of the fraction.

$f'' (\frac{7}{2}) = - \frac{e^{\frac{7}{2}}}{4}$

Step 7.2.10

The final answer is $- \frac{e^{\frac{7}{2}}}{4}$ .

$- \frac{e^{\frac{7}{2}}}{4}$

Step 7.3

At $\frac{7}{2}$ , the second derivative is $- 8.27886298$ . Since this is negative, the second derivative is decreasing on the interval $(3, 4)$

Decreasing on $(3, 4)$ since $f'' (x) < 0$

Step 8

Substitute a value from the interval $(4, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $4.1$ in the expression.

$f'' (4.1) = {(4.1)}^{2} e^{4.1} - 7 \cdot 4.1 \cdot e^{4.1} + 12 e^{4.1}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $4.1$ to the power of $2$ .

$f'' (4.1) = 16.81 e^{4.1} - 7 \cdot 4.1 \cdot e^{4.1} + 12 e^{4.1}$

Step 8.2.1.2

Multiply $16.81$ by $e^{4.1}$ .

$f'' (4.1) = 1014.32023451 - 7 \cdot 4.1 \cdot e^{4.1} + 12 e^{4.1}$

Step 8.2.1.3

Multiply $- 7$ by $4.1$ .

$f'' (4.1) = 1014.32023451 - 28.7 \cdot e^{4.1} + 12 e^{4.1}$

Step 8.2.1.4

Multiply $- 28.7$ by $e^{4.1}$ .

$f'' (4.1) = 1014.32023451 - 1731.76625404 + 12 e^{4.1}$

Step 8.2.2

Simplify by adding terms.

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Step 8.2.2.1

Subtract $1731.76625404$ from $1014.32023451$ .

$f'' (4.1) = - 717.44601953 + 12 e^{4.1}$

Step 8.2.2.2

Add $- 717.44601953$ and $12 e^{4.1}$ .

$f'' (4.1) = 6.63743163$

Step 8.2.3

The final answer is $6.63743163$ .

$6.63743163$

Step 8.3

At $4.1$ , the second derivative is $6.63743163$ . Since this is positive, the second derivative is increasing on the interval $(4, \infty)$ .

Increasing on $(4, \infty)$ since $f'' (x) > 0$

Step 9

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(3, 8 e^{3}), (4, 4 e^{4})$ .

$(3, 8 e^{3}), (4, 4 e^{4})$

Step 10